Groups

A group is a non-empty set G together with an operation * (called the group operation) that combines any two elements a and b of G to form an element of G, denoted by a * b, such that the following three properties hold:

1. Associativity: For all a, b, and c in G, (a * b) * c = a * (b * c).
2. Identity element: There exists an element e in G such that for every a in G, e * a = a * e = a.
3. Inverse element: For every a in G, there exists an element b in G such that a * b = b * a = e, where e is the identity element.

The identity element of a group is the element that, when combined with any other element of the group, leaves that element unchanged. In the group of integers under addition, the identity element is 0, because for any integer a, 0 + a = a + 0 = a.

Associativity is a property of groups that states that for any three elements a, b, and c in a group, the operation * is associative, meaning that (a * b) * c = a * (b * c).

The order of a group is the number of elements in the group. The group of integers under addition is infinite, meaning that it contains an infinite number of elements.

A cyclic group is a group that is generated by a single element. The group of integers under addition is cyclic, because it is generated by the element 1, meaning that every integer can be expressed as a sum of 1s.

The generator of a cyclic group is the element that generates the group. In the cyclic group of integers under addition, the generator is 1, because every integer can be expressed as a sum of 1s.

An abelian group is a group in which the operation * is commutative, meaning that for all a and b in the group, a * b = b * a. The group of quaternions under multiplication is non-abelian, because there exist quaternions a and b such that a * b ≠ b * a.

The order of a group is the number of elements in the group. The group of quaternions under multiplication has 8 elements.

A subgroup of a group G is a non-empty subset H of G that is itself a group under the same operation as G. The set of even integers under addition is a subgroup of the group of integers under addition, because it is a non-empty subset that is closed under addition and contains the identity element 0.

The order of a subgroup is the number of elements in the subgroup. The subgroup of the group of integers under addition consisting of the even integers is infinite, because it contains an infinite number of elements.

A normal subgroup of a group G is a subgroup H of G that is invariant under conjugation by every element of G. The set of even integers under addition is a normal subgroup of the group of integers under addition, because it is invariant under conjugation by every integer.

The quotient group of a group G by a subgroup H is the group G/H, which consists of the cosets of H in G. The quotient group of the group of integers under addition by the subgroup of even integers is the group of integers under addition, because the cosets of the subgroup of even integers are the sets of all integers that differ from each other by an even integer.

A group homomorphism is a function between two groups that preserves the group operation. The function that maps each integer to its absolute value is a group homomorphism from the group of integers under addition to the group of non-negative integers under addition, because it preserves the addition operation.

The kernel of a group homomorphism is the set of elements in the domain that are mapped to the identity element in the codomain. The kernel of the group homomorphism that maps each integer to its absolute value is the set of all non-negative integers, because the absolute value of every non-negative integer is the identity element 0.

A group isomorphism is a bijective group homomorphism. The function that maps each rational number to its reciprocal is a group isomorphism from the group of rational numbers under multiplication to the group of rational numbers under multiplication, because it is a bijective function that preserves the multiplication operation.